Hypergeometric functions are key

From Orthogonal Polynomials and Special Functions by Richard Askey:

At first the results we needed were in the literature but after a while we ran out of known results and had to learn something about special functions. This was a very unsettling experience for there were very few places to go to really learn about special functions. At least that is what we thought. Actually there were many, but the typical American graduate education which we had did not include anything about hypergeometric functions. And hypergeometric functions are the key to this subject, as I have found out after many years of fighting them.

Emphasis added.

Askey's book was written in 1975, and he was describing his experience from ten years before that. Special functions, and in particular hypergeometric functions, went from being common knowledge among mathematicians at the beginning of the 20th century to being arcane by mid century.

I learned little about special functions and nothing about hypergeometric functions as a graduate student. I first ran into hypergeometric functions reading in Concrete Mathematics how they are used in combinatorics and in calculating sums in closed form. Then when I started working in statistics I found that they are everywhere.

Hypergeometric functions are very useful, but not often taught anymore. Like a lot of useful mathematics, they fall between two stools. They're considered too advanced or arcane for the undergraduate curriculum, and not a hot enough area of research to be part of the graduate curriculum.

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